"We have got to be careful not to see what are ongoing and upgrading relationships here for Australia and the Pacific through those binary terms of the United States and China," Mr Morrison told reporters.
"They have their interests in the region, as do others.
"Our relationship with the Solomon Islands, our relationship with the Pacific, transcends all of that."
The United States has recently expressed concern that China - with whom it is locked in a trade war - has been trying to influence Pacific nations' diplomatic relationships with have with Taiwan.
Six Pacific island nations give diplomatic recognition to Taiwan, accounting for a third of its diplomatic allies.
But Chinese President Xi Jinping last week told the visiting prime minister of Vanuatu that his country is not seeking a sphere of influence among Pacific states.
Mr Morrison says it is entirely up to the Solomon Islands where it stands on Taiwan, which Beijing considers a sacred Chinese territory.
"They are an independent sovereign country who will make decisions in their national interests.
"It is not our place to provide advice or guidance on those decisions."
The prime minister announced funding to give Solomon Islanders an easier path to get work in Australia, and a bilateral infrastructure program worth up to $250 million in grant financing over 10 years.
"We're here because we are family with the Pacific," he said.
"I made it clear before the election that I wanted to make this a mark of our government that we were keenly interested at a very deep level about this relationship."
Mr Sogavare presented Mr Morrison with a ukelele and a traditional wood carving.
Having been formally welcomed at the airport, the Morrisons attended an informal barbecue at the home of the Australian high commissioner.
Mr Morrison next moves on to the UK to represent Australia at the 75th anniversary commemorations of the D-Day landings in Portsmouth.
He'll also catch up with world leaders including outgoing UK Prime Minister Theresa May, US President Donald Trump, French President Emmanuel Macron, German Chancellor Angela Merkel, and Canadian Prime Minister Justin Trudeau.
He will be hoping to build support for his proposed G20 initiative to stop terrorists exploiting social media, along with discussions on other topics including Iran, Syria, the global economy and world trade.
While in the UK, Mr Morrison will also address the Australia-UK Chamber of Commerce, underlining his government's commitment to the economic partnership with Britain and plans to move quickly on a free trade agreement once Brexit is resolved.
On the final leg of his trip in Singapore, he will join his counterpart Lee Hsien Loong for the annual leaders meeting, joined by Foreign Affairs Minister Marise Payne and Trade Minister Simon Birmingham.
At a lunch with some 400 businesspeople, Mr Morrison plans to highlight Australia's commitments to deepening economic and security ties with Southeast Asia.